Assuming the price function as p=90+2Q and cost function as c=10-0.5Q(2). Find the maximizing output and price.
Please solition
No
To find the maximizing output and price, we need to find the quantity at which the profit is maximized. Profit is calculated by subtracting the cost from the revenue.
Revenue (R) is calculated using the price function:
R = p * Q
Cost (C) is calculated using the cost function:
C = c * Q
Profit (P) is calculated by subtracting cost from revenue:
P = R - C
Given the price function p = 90 + 2Q and the cost function c = 10 - 0.5Q^2, we can substitute these equations into the profit equation.
R = (90 + 2Q) * Q
C = (10 - 0.5Q^2) * Q
P = R - C
P = (90 + 2Q) * Q - (10 - 0.5Q^2) * Q
Expanding and simplifying the equation, we get:
P = 90Q + 2Q^2 - 10Q + 0.5Q^3
To find the maximizing output, we need to find the quantity (Q) that gives the maximum value for profit. We can do this by taking the derivative of the profit function with respect to Q and setting it equal to zero.
dP/dQ = 90 + 4Q - 10 + 1.5Q^2
0 = 90 + 4Q - 10 + 1.5Q^2
Rearranging the equation, we get:
1.5Q^2 + 4Q - 10 = -90
Adding 90 to both sides, we get:
1.5Q^2 + 4Q - 10 + 90 = 0
Simplifying the equation, we get:
1.5Q^2 + 4Q + 80 = 0
To solve this quadratic equation, we can either factorize or use the quadratic formula. In this case, the quadratic formula is more appropriate:
Q = (-b ± sqrt(b^2 - 4ac)) / 2a
Using the values from the quadratic equation, a = 1.5, b = 4, and c = 80:
Q = (-4 ± sqrt(4^2 - 4 * 1.5 * 80)) / (2 * 1.5)
Calculating the square root, we get:
Q = (-4 ± sqrt(16 - 480)) / 3
Since the square root is negative, we have complex roots. Therefore, there is no valid positive quantity that maximizes profit.
In this case, there is no maximizing output and price.
To find the maximizing output and price, we need to find the quantity (Q) that maximizes the profit.
Profit (π) is calculated by subtracting the cost (c) from the revenue (R). Revenue can be determined by multiplying the price (p) with the quantity (Q). So, the profit can be expressed as:
π = R - c
Substituting the given price function (p) and cost function (c) into the profit equation, we get:
π = (90 + 2Q)Q - (10 - 0.5Q^2)
Expanding and rearranging the terms, we get:
π = 90Q + 2Q^2 - 10 + 0.5Q^2
Now, let's differentiate the profit equation with respect to Q to find the critical points where the profit is maximized:
dπ/dQ = 90 + 4Q + Q = 0
Combining like terms:
5Q + 90 = 0
Solving for Q:
5Q = -90
Q = -90/5
Q = -18
Since a negative quantity doesn't make sense in this context, we discard this solution.
Now, we can compute the second derivative to determine the nature of the critical point:
d²π/dQ² = 4
Since the second derivative is positive, it implies that the critical point we found is indeed a maximum point.
Thus, the maximizing output (Q) is 0, and to find the maximizing price, we substitute this value back into the price function:
p = 90 + 2(0)
p = 90
Therefore, the maximizing output is 0, and the corresponding maximizing price is 90.